Answer: TRUE. The firm can increase profits by increasing production.

Transcription

1 Solutions to 1.01 Make-up for Midterm Short uestions: 1. (TOTAL: 5 points) Explain whether each of the following statements is True or False. (Note: You will not get points for a correct answer without an explanation.) a) (5 points) In the short run, a competitive firm will never produce at levels where the marginal cost of production is decreasing (assume labor markets are perfectly competitive). Answer: TRUE. The firm can increase profits by increasing production. b) (5 points) If a production function has constant returns to scale, then the marginal cost is constant, both in the short-run and in the long-run. Answer: FALSE. Marginal cost is constant in the long-run but NOT in the short-run (because capital is fixed). For example, if production function is a CRS Cobb-Douglas, the marginal cost in the short-run is increasing. c) (5 points) A firm's short-run average cost curve is tangent to its long-run average cost curve only at a level of output where short-run marginal cost equals long-run marginal cost. (Use a diagram to explain your answer.) Answer: TRUE (See graph 1). d) (5 points) In an exchange economy with two agents, trade will take place only if the agents have different preferences. Answer: FALSE. They will trade if preferences are identical, but endowments are not. e) (5 points) If the production of the only goods in an economy is characterized by Constant Returns to Scale, then the marginal rate of transformation (MRT) is equal to the price ratio (of the two goods) at all points of the production possibilities frontier. Answer: TRUE, provided that the two goods are not perfect substitutes. If both goods are CRS, the MRT is a constant (PPF is linear). See graph.

2 Long uestions:. A firm producing widgets faces the following production function: KL Where K is capital and L is labor. The rental rate for capital is $ and wage rate for labor is $. (a) In the short-run, capital is fixed at units. Calculate the optimal factor demand for labor, and derive the short-run cost function of the firm. ) (.. ) ( * + + C r w C L L (b) Calculate the short-run average cost, average variable cost, average fixed cost, and marginal cost. Show these functions on a diagram. SRMC SRAFC SRAVC SRAC 8 + (c) In the long-run, all factors are variable. Calculate the optimal factor demands for labor and capital, and derive the long-run cost function of the firm. C K L L K L K r w MRTS KL 8 8 ) ( ; * * +

3 Now assume that there are 00 identical firms in a perfectly competitive industry, each facing the same production function as above. The market demand is represented by: D p (d) For the short-run, find the industry supply curve and the market price, as well as the quantity produced by each firm, and its profits. Will firms shut-down in the short-run? SRMC S 00q p Equilibrium : S S D 5p p p 8 00 q π ( p 8) > ( SRAVC ) So, no _ shutdown (e) Suppose the government imposes a per-unit tax of $5 on widgets. Find the new short-run equilibrium quantity and profits for each firm. Will firms shut-down in the short-run? SRAC q q SRAVC q + 5 SRAFC q SRMC 8q + 5 S S 5( SRMC 5) 5p 15 D 5p p p q π ( p 9) > ( SRAVC 7) So, no _ shutdown

4 UESTION 3: The sporting good sector in the Dominican Republic specializes in the production of baseball bats. The demand and supply of Dominican baseball bats in the US is given by: D( p) p S( p) 100 p 00 (a) Calculate the equilibrium price and quantity of Dominican baseball bats bought by US customers. Compute the consumer surplus for US customers and the producer surplus for Dominican firms (You may treat all Dominican firms as one single baseball bat producer). D S p 100p p p 8; q 600 Consumer Surplus 0.5*600*(1 8) 1800 Producer Surplus 0.5*600*(8 ) 1800 (b) The US Baseball Bat Producers Association (USBBPA) has lobbied the government for protection against Dominican producers. They have proposed that the government restrict the import of Dominican bats to 00, and that the government sell import licenses to the Dominican firm at a price R per Dominican bat imported. Calculate the value of R that will achieve the government s goal. S(p) 100(p R) 00 Also, p R R p 6 D S 100p R p 100p p p 100p 1000 p 10 R q 00 (c) Compute and graph the value of the new consumer surplus, producer surplus, and government revenue. What is the dead weight loss? What is the total US welfare? (Hint: Revenues and profits of the Dominican firm are remitted back to the Dominican Republic in their entirety.)

5 CS 0.5*00*(1 10) 800 PS 0.5*00*(6-) 800 Government Revenue 00*(10 6) 00 DWL 0.5*(600 00)*(10 6) 00 Total US Welfare CS + US Government Revenue (d) Suppose instead of imposing an import restriction, the US government requests the Dominican Republic government to sell export licenses for 00 bats to the Baseball firm. Assume that the Dominican government agrees, but sells these export licenses for free. Calculate and graph the new consumer surplus, producer surplus, dead weight loss, and total US welfare. US Welfare CS 800 PS DWL 00 (e) Now suppose instead of either of the above two policies, the Dominican firm reaches an agreement with the US government in which it pays the government a lump-sum amount of $00, which the government then redistributes to the member firms of USBBPA. Calculate and graph the new consumer surplus, producer surplus, dead weight loss, and total US welfare. CS 1800 PS DWL 0 US Welfare (f) Which of the above three policies is optimal for overall efficiency? Explain why? Policy 3 since DWL 0

6 5. (TOTAL: 5 points) Pati (C) and Jeff (J) live in an island, where only cheese (C) and Bread (B) are consumed. Pati enjoys cheese and bread only if eaten as a sandwich ( units of bread, one unit of cheese). Their utility functions (U) and initial endowments (E) are the following: Pati: Jeff: U P min C P, B p ; E P C E P B 6 U J 0.5 ln C J 0.5 ln B J ; E J C 3 E J B a) ( points) Draw the Edgeworth Box, labeling the axes and the coordinates of the four corners. Locate the endowment point and draw the indifference curves that go through the endowment point. Answer: See graph 3. b) ( points) The central planner of the island has a preference for Jeff and forces Pati to give Jeff two units of each good. Is the new allocation of bread and cheese Pareto Efficient? Explain. Answer: The allocation is NOT Pareto efficient, because at the new endowment point, we can still increase the utility of Jeff without decreasing the utility of Pati, if for example, she gives him more units of bread.

7 Now assume that there are many Patis and many Jeffs in this economy (the same number of each type), such that each agent must take the price of cheese and bread as given. To simplify the analysis assume P B 1. (Use the endowments given in part (a)). c) (10 points) Find the market equilibrium price for cheese (P C ). Find the equilibrium consumption bundles C P, B P, C J and B J. Represent the equilibrium in a separate Edgeworth Box below. What are the utility levels for any particular Patricia and Jeff at this equilibrium? Answer: Pati maximizes her utility by choosing (1) B P C P. Total endowment of bread implies that: () B P B J 10 and total endowment of cheese (multiplying by two both sides of the equation): (3) C P C J 10 Substracting (3) from () (and replacing (1)): () B J C J. Now,JeffmaximizeshisutilitybysettingMRS Price ratio: (5) B J C J P C Replacing () in (5) we find the equilibrium price of cheese: P C Pati s demand functions are given by: C P Pc 6.5 P C B P Pc 6 5 P C Using the total endowment constraint we have that: C J.5 B J 5 Utilities are given by: U P 5 U J 0.5Ln 1.5

8 Suppose the central planner in this economy is considering opening the economy to world trade. The world price of cheese is P C 1 while the world price of bread is P B 1. Opening the island to world trade would mean that: (1) All the traders in the island would take world prices as given (the island is a very small part of the world economy). () Each trader can buy as much cheese and bread as he can afford. (3) The total amounts of cheese and bread purchases are not limited by the initial endowments of the island's inhabitants. d) (7 points) Show that the Jeffs would not agree to open to trade, but the Patis would. If you were a central planner and only cared for the sum of their utilities, would you open the economy? Why or why not? (Hint: ln(1.5).506 and ln(1.5).53) Answer: At world prices Jeff will maximize his utility by buying: C J 0.5 IJ P C B J 0.5 IJ P B His utility will be : U J 0.5Ln 1.5 lower than before, so he is against opening to trade. On the other hand, Pati will buy: C P Pc 6 8 P C 3 B P 3 and her utility will be: U P 5 3 therefore, she agrees to free trade. Given that: U P U J (no free trade) 5 0.5ln U P U J (free trade) ln the central planner will decide to open the economy.

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